## Why Do We Live in Three Dimensions?

by Amy Shira Teitel[/caption]

Day to day life has made us all comfortable with 3 dimensions; we constantly interact with objects that have height, width, and depth. But why our universe has three spatial dimensions has been a problem for physicists, especially since the 3-dimensional universe isn’t easily explained within superstring theory or Big Bang cosmology. Recently, three researchers have come up with an explanation.

Most astronomers subscribe to Big Bang cosmology, the model that proposes that the universe was born from the explosion of an infinitely tiny point. The theory is supported by observations of the cosmic microwave background and the abundance of certain naturally occurring elements. But Big Bang cosmology is at odds with Einstein’s theory of general relativity – general relativity doesn’t allow for any situation in which the whole universe is one tiny point, which means this theory alone can’t explain the origin of the universe.

The incompatibility between general relativity and Big Bang cosmology has stumped cosmologists. But almost 40 years ago, superstring theory arose as a possible unifying theory of everything.

Superstring theory suggests that the four fundamental interactions among elementary particles – electromagnetic force, weak interaction, strong interaction, and gravity – are represented as various oscillation modes of very tiny strings. Because gravity is one of the fundamental forces, superstring theory includes an explanation of general relativity. The problem is, superstring theory predicts that there are 10 dimensions – 9 spatial and one temporal. How does this work with our 3 dimensional universe?

Superstring theory has remained little more than a theory for years. Investigations have been restricted to discussing models and scenarios since performing the actual calculations have been incredibly difficult. As such, superstring theory’s validity and usefulness have remained unclear.

But a group of three researchers, associate professor at KEK Jun Nishimura, associate professor at Shizuoka University Asato Tsuchiya, and project researcher at Osaka University Sang-Woo Kim, has succeeded in generating a model of the universe’s birth based on superstring theory.

Using a supercomputer, they found that at the moment of the Big Bang, the universe had 10 dimensions – 9 spatial and 1 temporal – but only 3 of these spatial dimensions expanded.

The team developed a method for calculating matrices that represent the interactions of strings. They used these matrices to calculate how 9 dimensional space changes over time. As they moved further back in time, they found that space is extended in 9 directions, but at one point only 3 directions start to expand rapidly.

In short, the 3 dimensional space that we live in can result from the 9 original spatial dimensions string theory predicts.

This result is only part of the solution to the space-time dimensionality puzzle, but it strongly supports the validity of superstring theory. It’s possible, though, that this new method of analyzing superstring theory with supercomputers will lead to its application towards solving other cosmological questions.

So would that mean that at the point of origin of the universe one would find the other 6 dimensions?

The 9 spatial dimensions become split into 3 space dimensions we observe plus 6 dimensions which are compactified into Calabi-Yau spaces. It would be as if a line in three space were really a “thickened” rope with an additional dimension corresponding to the radius of it. This is extended to even higher dimensions. The candidate I work on for this 6 dimensional space are a class of manifolds called K3 spaces (Kummer, Kahler and Kodaira). The internal motions in this Calabi-Yau space is what gives rise to the other forces in the universe.

This compactification is removed, or this wadded up space unfolds like some origami, when gravity unifies with the other forces. This occurs at the most early domain in the existence of the cosmology.

LC

I recall a previous comment discussion where the lack of granularity effects on light observed from distant sources seemed to be a problem for extra dimensions?

The lack of dispersion of light from distant sources indicates there are no influences from quantum fluctuations of light, or a discrete grid up of spacetime which violates Lorentz symmetry. These observations do put constraints upon physical theories. In particular it means that loop variable quantum gravity is either wrong or must be seriously modified. String theory is more flexible and its quantum of gravity is a graviton, which comes from the closed string. The graviton is something which has holographic content. These data indicate that quantum fluctuations of gravity can’t be something which involves stochastic ripples of space or spacetime in the way previously thought. These fluctuations must exist in some holographic context, such as in the fluctuation of event horizons or the graviton content in an anti-de Sitter spacetime which our spacetime is a boundary of.

Gravity is an odd ball of the forces in nature, for it is not really a force in the standard sense. Classical gravitation as given by general relativity does not describe motion according to acceleration, but rather as a system of local inertial frames. The recent development by Maldacena indicates that the boundary of an anti-de Sitter (AdS) spacetime is equivalent to a conformal field theory (CFT). The graviton is then something within the AdS which is determined by the holographic projection of the CFT on the surface. The symmetries of this CFT is contained in these compactified spaces. So quantum fluctuations of the CFT on the AdS boundary determine “gravity” in the AdS interior. Further, local gravity we observe (living on the AdS boundary) is some symmetry breaking of the CFT (the occurrence of masses etc) which involves spacetime fluctuations of the event horizon, but maybe not the spacetime itself.

LC

Not discerning an obvious yes or no from all that 🙂

The answer, if there is an answer in this business, is that this dispersion is not a necessary aspect of all these theories. This data constrains theories, where a good number of them are eliminated.

LC

I guess it’s way to early for an obvious yes or no. A definite maybe — that may be possible. 😉

In other words everywhere in the Universe.

I’ve found two of their papers:

Expanding (3+1)-dimensional universe from a Lorentzian matrix model for superstring theory in (9+1)-dimensions;

Expanding universe as a classical solution in the Lorentzian matrix model for nonperturbative superstring theory.

But only in the lay-man use of the term “theory”. String theory is not a valid theory in the scientific sense. Until now it is still and only a mathematical toy (as far as I know) and has not produced any measurable results making possible its falsification.

I think that even this described result cannot be “observed”. But if I’m wrong, please correct me. 😉

That is what is claimed by its detractors, but it is trivially false.

Already its conception was promoted by its ability to predict the “flux tubes” that pops up between particles in models of the strong force. So it is a predictive theory and have produced measurable, even testable observations which it has tested successfully on. This was in ’73, I believe.

String theory also predicts black hole entropy correctly. And it is predictive in the Planck regime, predicting objects of strings and branes, which has been started to be probed recently. (By supernova photon timing and polarization observations.)

The problem is that string theory was king for a year. The year after the successful prediction of flux tubes QCD came along and predicted it by far simpler and more natural means. And as for the black hole results it wasn’t even first, simpler semiclassical approximations allowed for it outright from known physics.

The Planck regime results have not been accepted by many or most, and have little bearing on string theory as of yet. Though those results recurring insistence on an absence of small granular string manifold dimensions seems to this layman to be a pressing problem.

String theory may be fine physics, but it doesn’t look to me at the moment that it will be the Ultimate Answer to the Ultimate Question of Life, The Universe, and Everything“.

Let us be honest about this. Is string theory somehow the final and complete theory of the universe? No, or at least I would be surprised if it were. However, I think there are stringy aspects to the universe, where some signatures of this have been found. A part of the problem is that the mathematical foundations of this are so difficult that we are forced to make lots of simplifying assumptions. Newton when he was working on his mechanics worked out circular orbits, which are approximations that simplify things considerably and permit you to look at central issues. We have similar issues today, where the algebraic geometry of string theory is a part of something almost unimaginably difficult — maybe intractable.

There is this competing theory called loop quantum gravity. It is in many ways a very direct approach to quantum gravity that has Einstein’s general theory of relativity as its kernel. Yet quantum versions of spacetime connections violate Lorentz symmetry near the Planck scale, and they have action or a unit of spin which makes it difficult to construct classical gravity or compute the entropy of a black hole. Yet I don’t think loop variable theory is completely wrong.

String theory most naturally describes inflationary cosmology, which with anisotropy measurements appears to be realistic or has a modicum of empirical support. String theory did emerge from hadron physics, and the quark-gluon system of QCD is a stringy physics. The stringy structure of the universe appears in a low energy structure. Physical structures do seem to emerge in different guises, such as supersymmetry has been found as an emergent symmetry in low energy nuclear physics. The AdS~CFT correspondence appears in the context of superconductivity with heavy metals.

LC

If we should be completely honest, I think the LQG is considered completely wrong by the physicist side of the theoretical physics divide, while mathematicians may yearn for it.

The reasons, as I understand it, is:

– LQG breaks Lorentz invariance, as you noted on this symmetry.

– LQG has no energy gap setting a lowest energy resulting in an action that follows an action principle, as you noted on the action.

– Related to the latter point, LQG gives no dynamics! You can’t even construct an idealized harmonic oscillator based on its structure, so it lacks the fundamental degrees of freedom which makes up the physics we see.

Those seemingly unsurmountable problems are disregarded by the LQG community, which persists in putting out arxiv papers promising all these things and never delivering. (Yes, I am bored by reading all so called “proofs” for these claims. They never are, unless you squint and put in enough fairies.)

The failures of loop variable theory are what make it beautiful! That might sound odd, but really, it is the minimal route to quantum gravity. It makes far fewer of the conjectures seen in string theory, which is really closer to elementary particle theory, and sticks close to Einstein’s theory of relativity. Yet the theory is plagued by a host of problems which make it almost unworkable. This gives us the most beautiful there is in science — why? Questions are always more interesting than old established answers.

Loop variables are based on the ADM approach to classical gravity. This has no time, and defines constraints NH = 0 and N^iH_i = 0. The Hamiltonian is g^{-1/2}(TrK^2 – (TrK)^2 – R^{(3)}, K_{ij} = extrinsic curvature of spatial manifold, which just defines a spatial manifold in a system of such manifolds under a diffeomorphism described by a lapse function. This is quantized under canonical methods which leads to LQG. The theory has no time variable, and Barbour makes a huge point of this in his proclamation “Time is an illusion.” This is why there is no dynamics.

There is an alternative theory outlined by Banos, Teitelboim and Zanelli (BTZ) which has 2 space and 1 time dimension. Now suppose there is a noncommutative geometry which tells us that

?r?t ~ G?/c^3,

there is then a noncommutative quantization between the two representations.

The additional dimension, which is time in the ADM and space in the BTZ, emerges by the holographic principle. Another way to think of this is that the spinor group for the ADM is SU(2) ~ SO(3), which are just the rotations in space. In BTZ this group is SU(1,1), which is a group of Lorentz boosts. The complementarity gives a complete group SU(2)xSU(1,1) ~ SO(3,1) ~ SL(2,C)

LC

Sir, as a matter of fact, it is string theory, which might actually lead to the grand unification theory, and when that is known, we may know, as Einstein said, “The mind of God”. String theory is mathematically perfect in the sense that it doesn’t violate any known physical laws. Also, it accurately explains the quantum phenomenon of the minutest particles, as well as predicting them. but the real trouble is that, they haven’t been experimentally verified, as the energy needed is of the planck scale, which we can’t create at the present. if it happens, then we can know what happened during inflation, that is, the Big bang. M-theory can then be verified. Till then we have to wait..

Knots can only exist in 3-dimensions. They are essential to life, at least as we know it (proteins). So, we can only exist in a 3-dimensional universe and, I suspect, can only observe the universe we exist in.

The KEK mathematicians have only shown that a string model is consistent with one aspect of our 3-D universe. If we can’t observe anything else, it’s math not science.

Knots exist for very subtle reasons. The standard knot is a one dimensional curve which threads through three dimensions. It is more technically due to something called a Skein relationship with a polynomial expression. This is the Jones polynomial, which has been generalized into something called the HOMFLY polynomial

This goes back to something called the Hopf fibration of spheres. The elementary one is

S^1 — > S^3 — > S^2

The S^1 is any curve, which is a closed loop. In this fibration there is a “thickening” of the two dimensional S^2 which allows this curve to pass “right over left” or “left over right” or not intersect. This relationship defines a term in a path integral called a Wilson loop. This is a knot invariant which is connected to the theory of complex variables. The S^3 in the middle feeds into another Hopf fibration

S^3 — > S^7 — > S^4.

This also defines a skein relation! Here we are knotting tree dimensional spheres in seven dimensions. The Skein relations define a Chern-Simons invariant 4?k = ?? for ? = ?/d? + (2/3)?/?/? a 3-form measure on a 3-sphere. This is an invariant of the set of quaternions. The topology of this is connected to Milnor’s work on inequivalent diffeomorphisms on the 7-sphere. It gets better! The next level is the Hopf fibration

S^7 — > S^{15} — > S^8

where this is a Skein relation on 7-spheres in 15 dimensions! This corresponds to the octonions.

The first level of these Hopf fibrations is S^0 — > S^1 — > S^1. You might notice a pattern with the dimensions here; the middle sphere has dimension equal to the sum of the dimensions of other two, and the two on the left and right differ by one dimension. This first defines the algebra of real numbers, which has the following properties:

Betweeness

Commutativity

Associativity

The next fibration for complex numbers removes the betweeness property — two points on two dimensions have no point between then in common, for a whole class of curves can connect them. The quaternions are not commutative, and finally the octonions are not associative; in other words a(bc) – (ab)c is not zero in general. The right dimension also have 1, 2, 4, and 8, or dim = 2^n, for n = 0, 1, 2, 3. This has certain graph representations in the Fano plane, which is a discrete system from the Moufang or projective Cayley plane.

Can one go higher with

S^{15} — > S^{31} — > S^{16}?

These are the sedenions, but they have no division algebra. We run out of mathematics and enter “insanity.” These have certain Lie algebraic correspondences, and higher systems do exist, but they are not extensions of the Hopf fibrations. A continuation of the Hopf fibration leads to something called Bott periodicity. The larger systems from a group theory or algebraic approach correspond to sporadic groups, which are more bizarre than the exceptional groups, such as E_8 which corresponds to the octonions. Further, the these sporadic groups form automorphisms of the Fisher-Griess group — sometimes called the monster group with a prime number factorization which gives its dimension as about 8×10^{51} dimensions.

All of this has a bearing on this physics — even the monster group!

LC

So I guess these 6 rather small dimensions might explain why the observable 3D universe is rather uniform across vast 3D distances? Every point in “our” 3D universe must be closely connected somehow, or how else would you explain the similarity of everything everywhere? (Perhaps some kind of “inertia” in laws of nature and physical constants, but wouldn’t that also call for an extra dimension where that inertia “lives”??)

Well just a layman musing here, I hope someone will take the bait and expand/explain this weird similarity phenomenom a bit further..

IF I understand the question. One controversial viewpoint reply: Why the uniformity through time, and similarity in space? – A common origin: Universal frame of laws enforce throughout the Cosmos ( producing the same results everywhere ), similarity in architectural features and forms reflecting a common design-imprint: the entire vast Complex made from the same precisely scaled, highly structured Blueprint.

Our 3-dimensional time-space may be intimately “connected” to a higher plane reality, one not subject to its 3D limitations, or its matter-distance time-measures, but interfacing with it – the origin of it? – from a greater level of existence: Like the relationship between a substantial, imposing granite pillar, and its ephemeral shadow – one endures, the other changes, and fades.

I like that idea of the additional dimensions providing the physical controls that we observe in our three. We don’t see a reason the speed of light is what it is, why isn’t it ten times as fast? What limits it? Perhaps one of the other six dimensions has a controlling factor, a barrier. If we could peek into that other dimension, maybe we could manipulate it and ‘break the light barrier’. Interstellar travel, here we come.

So the other dimensions could provide the underlying framework of our physical laws. Let’s speculate that one dimension is associated with the electromagnet force. The ‘depth’ of that dimension could define the strength of the force.

Lots of problems with this concept, of course, but I like the possibilities it leads to.

“…the universe was born from the explosion of an infinitely tiny point.”

A common misconception I long subscribed to ——–>

—————————————————————————

“Was the Big Bang an explosion”?

“No, the Big Bang was not an explosion. We don’t know what, exactly, happened in the earliest times, but it was not an explosion in the usual way that people picture explosions. There was not a bunch of debris that sprang out, whizzing out into the surrounding space. In fact, there was no surrounding space. There was no debris strewn outwards. Space itself has been stretching and carrying material with it.”

– WAMP (Wilkinson Microwave Anisotropy Probe), NASA —> ttp://map.gsfc.nasa.gov/site/faq.html

—————————————————————————–

“Big Bang Theory – Common Misconceptions”:

“There are many misconceptions surrounding the Big Bang theory. For example, we tend to imagine a giant explosion. Experts however say that there was no explosion; there was (and continues to be) an expansion. Rather than imagining a balloon popping and releasing its contents, imagine a balloon expanding: an infinitesimally small balloon expanding to the size of our current universe.”

– All About Science —> http://big-bang-theory.com/

Now, why would anybody have a misconception about a Theory titled: The BIG BANG?

—————————————————————————–

“But Big Bang cosmology is at odds with Einstein’s theory of general relativity – general relativity doesn’t allow for any situation in which the whole universe is one tiny point.”

Well, could it be the model is correct – except for the “singularity” point of origin? Model another “point” of origin, where it is NOT required to have colossal volume and density infinitely compacted together into a hypothetical beginning, but rather allowing for a sudden flash of Universe Creation from source: formation through an instantaneous expanding creation – actual Creation as opposed to transformation, as in infinitesimal “singularity” becomes infinitely huge Cosmos – inflating out and forming the enormous organized complex of law-governed Space, and its ordered energy-matter formations of Time?

(String Theory may be perfectly compatible with the alternate Model, I just do not understand it enough.)

Perhaps Fred Hoyle has had the last laugh as “Big Bang” was originally a disparaging term for the idea he hated, by causing so many people’s misconceptions of it. Perhaps cosmologists should run a competition to find a new more correct name for the creation event, perhaps with an apt acronym as a bonus.

The steady state cosmology model of Hoyle, Bondi, Gold and Narlikar is dead as a doornail. If completely fails to match observational data. Further, if the universe were steady state something curious would happen. The smallest perturbation would cause it to either expand or collapse. This perturbation could be a quantum fluctuation.

LC

It is amazing to contemplate, all the complexity that rises from “nothing”.

How the universe emerged is still a work in progress. Inflation occurred at 10^{-33} seconds into the existence of the universe for a very brief period of 10^{-35} seconds. This is very close to the origin, but it is 10^{10} Planck units of time after the universe emerged. How the universe emerged and the physics behind it are not well understood. Steinhardt’s ekpyrotic model involves colliding branes, which invokes some pre-existing system. Smolin thinks that cosmologies quantum mechanically tunnel into existence by the vacuum polarization near singularities of black hole, where these black holes exist in other cosmologies within the multiverse setting.

General relativity does not describe topology changes. The universe may have emerged from a point, or some small spatial sphere that transformed into a flat space, but general relativity works with a geometry that maintains a constant topology. Geometry describes the gridded map of a space, say a map a surveyor works on land, but the topology is fixed. The physics of how a cosmology, or a spacetime, emerges may involve topology changes.

The question about why space is three dimensional probably has something to do with this. It might have something to do with the Hopf fibration issues I outlined below. There might also be some duality between 3 dimensional space and 2+1 dimensions of space plus time. These are very interesting problems to think about, and they do push at the foundations of our understanding of physics and cosmology.

LC

That is one tall order with at least 3 ad hocs.

– First you have to append an ad hoc big bang singularity to standard cosmology. All you need for eternal inflation is an inflation potential with a local minimum (Bousso).

Also, let us not forget that the quantum creation of a universe may presently be in trouble. Or not, I haven’t heard of a reaction to that result.

But if it is true and tunneling is forbidden, you may have entered a dead end: how do you predict a cosmological singularity in the first place?

– Then you need to introduce ad hoc IR and UV (I think) cutoffs in the model of Kim et al, according to the papers that Ivan3man linked to. (“Gaussian expansion method”.) They can remove at least one in the low energy limit, but presumably they are still needed in their early time simulations.

This is in order to replace the naturally stable and accepted method of Wick rotations.

So, yes, possible, but perhaps not likely and definitely not natural in the context.

This is common claim even among physicists, but I think it amounts to “folk science” as good hypotheses has been around for a very long time:

– In an anthropic (say, eternal inflation) universe, we need to have 3 spatial dimensions to have bags that can contain insides isolated from the environment (cells). Sankey raises another problem for non-3D biochemistry, proteins folds needs to be stable to have polymer structural building blocks or catalysts.

– In a multiverse (say, from eternal inflation), we need to have 1 time dimension and 3 spatial dimensions to have interesting physics.

Tegmark:

“With more or less than one time-dimension, the partial differential equations of nature would lack the hyperbolicity property that enables observers to make predictions. In a space with more than three dimensions, there can be no traditional atoms and perhaps no stable structures. A space with less than three dimensions allows no gravitational force and may be too simple and barren to contain observers.”

Look at his figure to see how unpredictability cuts off too many or too few time dimensions and simplicity or complexity (instabilities) cuts off too many or too few space dimensions.

I will add this tentative one, though it is a personal observation:

– It is only in 4D (1+3 D) that there is an interesting (actually infinite, I think) collection of manifolds. Presumably that is why physics over manifolds with freedom (say, string theory) ends up promoting 3 spatial dimensions.

Maybe lcrowell can weigh in on that one.

That doesn’t seem to pass the smell test.

General relativity (GR) suggests and handles singularities, say in black holes. Singularities are nice, at least for those who looks for Theories Of Everything (TOEs), since they provide an infinite set of degrees of freedom for free. So a TOE theory that wants to bind them to a unique parameter set has one less problem to worry about.

What GR doesn’t do is predict their quantum mechanical behavior. (Well, duh.) It isn’t a complete physics in that sense.

Not being a physicist, and not understanding the math, I’m almost afraid to post.

First question please: I heard of a theory called branes. I like to watch the Science Channel.

I heard one theory predicts our universe is the result of a collision of two branes.

For those who think branes exist, do we know where, in the collision, we are?

I had the impression, from the science program, the theory is, the collision happened,

and we are the result. I was wondering if the collision could still be happening.

I heard people believe the rate of expansion of the universe is increasing.

I am looking for an explanation why the rate of of expansion is increasing.

Second question please: I watched a science program about Dark Matter and Dark Energy.

I have no idea if these things exist. Theoretically, in which dimensions do they exist?

Third question please: If Dark Matter and Dark Energy exist, where do they come from?

I was wondering if a Black Hole might produce Dark Energy.

I was wondering if Dark Energy is really another state of Dark Matter…silly me.

I was thinking Dark Energy might be the “gaseous” state of Dark Matter,

Dark Matter the “liquid” state, and …, what could be the solid state.

This question is hair-brain anyway.

If you watch the Science Channel you need only one caveat. Be skeptical of anything Michio Kaku says.

If you Google any of those topics, you will have enough reading material to last a lifetime. Enjoy 🙂

D-branes are surfaces where the endpoints of strings are attached, and the those ends satisfy Dirichlet boundary conditions. This means that in the center of mass frame of the string the endpoints are fixed. There are Newman conditions, defining N-branes, where momentum of the endpoints is fixed. Branes are curious, and as it turns out they are not quantum mechanical. They are composed of many quantum numbers and are then really classical. If one where to look closely on a small region of a brane you find they are composed of strings. As such they are really a defect in the vacuum analogous to an electron surface in solid state physics. There is a dual element to this as well, for in an S-duality a string can be composed of D0-branes (D-branes of zero dimension which are like point particles), making a string similar to beads on a string.

The Steinhardt ekpyrotic theory invokes collisions between branes, where two branes might be connected by open strings with endpoints on them. The collisions between them reset the energy on the branes and this corresponds to a big bang.

You might want to read some on dark energy, for this is some physics associated with the vacuum energy. Dark matter is some form of quantum field particle which does not interact by the electromagnetic field. This is what makes them “dark.” These could be supersymmetric partners of known elementary particles.

LC

Seems to me “three dimensions” is simply a mathematical notion, defined as three directions which are separated by 90 degree angles. The 90 degrees mean that these dimensions are independent and that one can vary without effecting the others. This strikes me as a cultural construction, not an insight into reality. Why? Because there are many parallel “directions” in which the properties of objects can vary without effecting the others. What about colour or orientation for example?

I suppose the physicist would say that orientation is reduced to 3 dimensional movement if the object is considered a group of related objects moving through space, and the same for colour as the bouncing of light rays off of objects. So the solution seems to be to break objects into smaller and smaller units (sound familiar?) where these smaller and smaller units have nothing but positions in space and time. Regarding 9 dimensional space, it seems obvious that 9 is the number of dimensions required to satisfy the mathematical models to explain the phenomena in question. So how many dimensions are there really? Well, it depends on the level of abstraction in the model/description.

https://plus.google.com/104335933112525967488/posts/PkjSnEeXDev

What I wonder is this… If, at some point, 3 dimensions expanded enough to be what we sense today… Might it be possible, since expansion is still continuing, that the universe reaches a point where another dimension starts to expand… What would be the effect? Would we even be able to preceive it? How would that change the “laws” of physics?

It should be obvious, if the focus was more on physics than mathematics, that gravitation is not a force. Gravitation is a distortion of spacetime geometry in the presence of mass. The only time gravitation and force coincide is when gravitation is interrupted, as at or below the surface of a large mass surrounding a geometric vortex.